1. APPLICATIONS OF DIFFERENTIATION 4

2. EXTREME VALUE THEOREM If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b]. Theorem 3

3. FERMAT’S THEOREM If f has a local maximum or minimum at c, and if f ’(c) exists, then f ’(c) = 0. Theorem 4

4. 4.1 Maximum and Minimum Values In this section, we will learn: How to find the maximum and minimum values of a function. APPLICATIONS OF DIFFERENTIATION

5. OPTIMIZATION PROBLEMS Some of the most important applications of differential calculus are optimization problems.  In these, we are required to find the optimal (best) way of doing something.

6. EXAMPLES Here are some examples of such problems that we will solve in this chapter.  What is the shape of a can that minimizes manufacturing costs?  What is the maximum acceleration of a space shuttle? (This is an important question to the astronauts who have to withstand the effects of acceleration.)

7. These problems can be reduced to finding the maximum or minimum values of a function.  Let’s first explain exactly what we mean by maximum and minimum values. OPTIMIZATION PROBLEMS

8. MAXIMUM & MINIMUM VALUES A function f has an absolute maximum (or global maximum) at c if f(c) ≥ f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Definition 1

9. MAXIMUM & MINIMUM VALUES Similarly, f has an absolute minimum at c if f(c) ≤ f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are called the extreme values of f. Definition 1

10. MAXIMUM & MINIMUM VALUES The figure shows the graph of a function f with absolute maximum at d and absolute minimum at a.  Note that (d, f(d)) is the highest point on the graph and (a, f(a)) is the lowest point.

11. LOCAL MAXIMUM VALUE If we consider only values of x near b—for instance, if we restrict our attention to the interval (a, c)—then f(b) is the largest of those values of f(x).  It is called a local maximum value of f.

12. Likewise, f(c) is called a local minimum value of f because f(c) ≤ f(x) for x near c—for instance, in the interval (b, d).  The function f also has a local minimum at e. LOCAL MINIMUM VALUE

13. MAXIMUM & MINIMUM VALUES In general, we have the following definition. A function f has a local maximum (or relative maximum) at c if f(c) ≥ f(x) when x is near c.  This means that f(c) ≥ f(x) for all x in some open interval containing c.  Similarly, f has a local minimum at c if f(c) ≤ f(x) when x is near c. Definition 2

14. MAXIMUM & MINIMUM VALUES The function f(x) = cos x takes on its (local and absolute) maximum value of 1 infinitely many times—since cos 2nπ = 1 for any integer n and -1 ≤ cos x ≤ 1 for all x. Likewise, cos (2n + 1)π = -1 is its minimum value—where n is any integer. Example 1

15. MAXIMUM & MINIMUM VALUES If f(x) = x 2, then f(x) ≥ f(0) because x 2 ≥ 0 for all x.  Therefore, f(0) = 0 is the absolute (and local) minimum value of f. Example 2

16. MAXIMUM & MINIMUM VALUES This corresponds to the fact that the origin is the lowest point on the parabola y = x 2.  However, there is no highest point on the parabola.  So, this function has no maximum value. Example 2

17. MAXIMUM & MINIMUM VALUES From the graph of the function f(x) = x 3, we see that this function has neither an absolute maximum value nor an absolute minimum value.  In fact, it has no local extreme values either. Example 3

18. MAXIMUM & MINIMUM VALUES The graph of the function f(x) = 3x 4 – 16x 3 + 18x 2 -1 ≤ x ≤ 4 is shown here. Example 4

19. MAXIMUM & MINIMUM VALUES You can see that f(1) = 5 is a local maximum, whereas the absolute maximum is f(-1) = 37.  This absolute maximum is not a local maximum because it occurs at an endpoint. Example 4

20. MAXIMUM & MINIMUM VALUES Also, f(0) = 0 is a local minimum and f(3) = -27 is both a local and an absolute minimum.  Note that f has neither a local nor an absolute maximum at x = 4. Example 4

21. MAXIMUM & MINIMUM VALUES We have seen that some functions have extreme values, whereas others do not. The following theorem gives conditions under which a function is guaranteed to possess extreme values.

22. EXTREME VALUE THEOREM If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b]. Theorem 3

23. The theorem is illustrated in the figures.  Note that an extreme value can be taken on more than once. EXTREME VALUE THEOREM

24. Although the theorem is intuitively very plausible, it is difficult to prove and so we omit the proof. EXTREME VALUE THEOREM

25. The figures show that a function need not possess extreme values if either hypothesis (continuity or closed interval) is omitted from the theorem. EXTREME VALUE THEOREM

26. The function f whose graph is shown is defined on the closed interval [0, 2] but has no maximum value.  Notice that the range of f is [0, 3).  The function takes on values arbitrarily close to 3, but never actually attains the value 3. EXTREME VALUE THEOREM

27. This does not contradict the theorem because f is not continuous.  Nonetheless, a discontinuous function could have maximum and minimum values. EXTREME VALUE THEOREM

28. The function g shown here is continuous on the open interval (0, 2) but has neither a maximum nor a minimum value.  The range of g is (1, ∞).  The function takes on arbitrarily large values.  This does not contradict the theorem because the interval (0, 2) is not closed. EXTREME VALUE THEOREM

29. The theorem says that a continuous function on a closed interval has a maximum value and a minimum value. However, it does not tell us how to find these extreme values.  We start by looking for local extreme values. EXTREME VALUE THEOREM

30. LOCAL EXTREME VALUES The figure shows the graph of a function f with a local maximum at c and a local minimum at d.

31. It appears that, at the maximum and minimum points, the tangent lines are horizontal and therefore each has slope 0. LOCAL EXTREME VALUES

32. We know that the derivative is the slope of the tangent line.  So, it appears that f ’(c) = 0 and f ’(d) = 0. LOCAL EXTREME VALUES

33. The following theorem says that this is always true for differentiable functions. LOCAL EXTREME VALUES

34. FERMAT’S THEOREM If f has a local maximum or minimum at c, and if f ’(c) exists, then f ’(c) = 0. Theorem 4

35. The following examples caution us against reading too much into the theorem.  We can’t expect to locate extreme values simply by setting f ’(x) = 0 and solving for x. FERMAT’S THEOREM

36. If f(x) = x 3, then f ’(x) = 3x 2, so f ’(0) = 0.  However, f has no maximum or minimum at 0—as you can see from the graph.  Alternatively, observe that x 3 > 0 for x > 0 but x 3 < 0 for x < 0. Example 5 FERMAT’S THEOREM

37. The fact that f ’(0) = 0 simply means that the curve y = x 3 has a horizontal tangent at (0, 0).  Instead of having a maximum or minimum at (0, 0), the curve crosses its horizontal tangent there. Example 5 FERMAT’S THEOREM

38. The function f(x) = |x| has its (local and absolute) minimum value at 0.  However, that value can’t be found by setting f ’(x) = 0.  This is because—as shown in Example 5 in Section 2.8—f ’(0) does not exist. Example 6 FERMAT’S THEOREM

39. WARNING Examples 5 and 6 show that we must be careful when using the theorem.  Example 5 demonstrates that, even when f ’(c) = 0, there need not be a maximum or minimum at c.  In other words, the converse of the theorem is false in general.  Furthermore, there may be an extreme value even when f ’(c) does not exist (as in Example 6).

40. The theorem does suggest that we should at least start looking for extreme values of f at the numbers c where either:  f ’(c) = 0  f ’(c) does not exist FERMAT’S THEOREM

41. Such numbers are given a special name—critical numbers. FERMAT’S THEOREM

42. CRITICAL NUMBERS A critical number of a function f is a number c in the domain of f such that either f ’(c) = 0 or f ’(c) does not exist. Definition 6

43. Find the critical numbers of f(x) = x 3/5 (4 - x).  The Product Rule gives: Example 7 CRITICAL NUMBERS

44.  The same result could be obtained by first writing f(x) = 4x 3/5 – x 8/5.  Therefore, f ’(x) = 0 if 12 – 8x = 0.  That is, x =, and f ’(x) does not exist when x = 0.  Thus, the critical numbers are and 0. Example 7 CRITICAL NUMBERS

45. In terms of critical numbers, Fermat’s Theorem can be rephrased as follows (compare Definition 6 with Theorem 4). CRITICAL NUMBERS

46. If f has a local maximum or minimum at c, then c is a critical number of f. CRITICAL NUMBERS Theorem 7

47. CLOSED INTERVALS To find an absolute maximum or minimum of a continuous function on a closed interval, we note that either:  It is local (in which case, it occurs at a critical number by Theorem 7).  It occurs at an endpoint of the interval.

48. Therefore, the following three-step procedure always works. CLOSED INTERVALS

49. CLOSED INTERVAL METHOD To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: 1.Find the values of f at the critical numbers of f in (a, b). 2.Find the values of f at the endpoints of the interval. 3.The largest value from 1 and 2 is the absolute maximum value. The smallest is the absolute minimum value.

50. CLOSED INTERVAL METHOD Find the absolute maximum and minimum values of the function f(x) = x 3 – 3x 2 + 1 -½ ≤ x ≤ 4 Example 8

51. As f is continuous on [-½, 4], we can use the Closed Interval Method: f(x) = x 3 – 3x 2 + 1 f ’(x) = 3x 2 – 6x = 3x(x – 2) CLOSED INTERVAL METHOD Example 8

52. CLOSED INTERVAL METHOD As f ’(x) exists for all x, the only critical numbers of f occur when f ’(x) = 0, that is, x = 0 or x = 2. Notice that each of these numbers lies in the interval (-½, 4). Example 8

53. CLOSED INTERVAL METHOD The values of f at these critical numbers are:f(0) = 1 f(2) = -3 The values of f at the endpoints of the interval are:f(-½) = 1/8 f(4) = 17  Comparing these four numbers, we see that the absolute maximum value is f(4) = 17 and the absolute minimum value is f(2) = -3. Example 8

54. CLOSED INTERVAL METHOD Note that the absolute maximum occurs at an endpoint, whereas the absolute minimum occurs at a critical number. Example 8

55. CLOSED INTERVAL METHOD The graph of f is sketched here. Example 8

56. Use calculus to find the exact minimum and maximum values. f(x) = x – 2 sin x, 0 ≤ x ≤ 2π. Example 9 EXACT VALUES

57. The function f(x) = x – 2 sin x is continuous on [0, 2π]. As f ’(x) = 1 – 2 cos x, we have f ’(x) = 0 when cos x = ½.  This occurs when x = π/3 or 5π/3. Example 9 b EXACT VALUES

58. EXTREME VALUE THEOREM If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b]. Theorem 3

59. FERMAT’S THEOREM If f has a local maximum or minimum at c, and if f ’(c) exists, then f ’(c) = 0. Theorem 4

60. CRITICAL NUMBERS A critical number of a function f is a number c in the domain of f such that either f ’(c) = 0 or f ’(c) does not exist. Definition 6

61. CLOSED INTERVAL METHOD To find the absolute maximum and minimum values of a continuous function f on a closed interval [a, b]: 1.Find the values of f at the critical numbers of f in (a, b). 2.Find the values of f at the endpoints of the interval. 3.The largest value from 1 and 2 is the absolute maximum value. The smallest is the absolute minimum value.